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Thread: difficult sin x function

  1. #1
    Senior Member pacman's Avatar
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    difficult sin x function

    Solve for x: (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27
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  2. #2
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    Quote Originally Posted by pacman View Post
    Solve for x: $\displaystyle (16/81)^{\sin^2 x} + (8/9)^{\cos^2 x} = 26/27$
    Something wrong here? If you put $\displaystyle y = \sin^2x$ then $\displaystyle 0\leqslant y\leqslant1$. But the function $\displaystyle (16/81)^y + (8/9)^{1-y}$ is greater than 1 throughout the interval [0,1], so it can never be equal to 26/27.
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  3. #3
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    ok say instead of 26/27 it was 3/2.

    how can you solve it for an exact soln? w/o using any approximating methods
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  4. #4
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    Quote Originally Posted by Krahl View Post
    ok say instead of 26/27 it was 3/2.

    how can you solve it for an exact soln? w/o using any approximating methods

    If It is $\displaystyle ( \frac{4}{9})^{\cos^2{x}} $ not

    $\displaystyle ( \frac{8}{9})^{\cos^2{x}} $

    Sub $\displaystyle (\frac{4}{9})^{\sin^2{x}} = u $then

    $\displaystyle
    u^2 + (4/9) u^{-1} = M
    $

    $\displaystyle u^3 - M u + (4/9) = 0 $

    It becomes a cubic equation

    after solving u $\displaystyle u = (\frac{4}{9})^{\sin^2{x}} $

    $\displaystyle x = \arcsin{ \sqrt{ \frac{ \ln{u} }{ \ln(4/9) } } }$
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  5. #5
    Senior Member pacman's Avatar
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    i plot this one: y = (16/81)^(sin^2 x) + (8/9)^(cos^2 x) for x =0 to pi radian. In my graph it looks like this, any comments?
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  6. #6
    Senior Member pacman's Avatar
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    and this one, (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27; what does it mean?
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  7. #7
    Super Member Failure's Avatar
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    Quote Originally Posted by pacman View Post
    i plot this one: y = (16/81)^(sin^2 x) + (8/9)^(cos^2 x) for x =0 to pi radian. In my graph it looks like this, any comments?
    My comment: it is not true that this is the plot of $\displaystyle y = (16/81)^{\sin^2 x} + (8/9)^{\cos^2 x}$. Rather, this is the plot of $\displaystyle y = (16/81)^{sin^2 x} + (8/9)^{cos^2 x}-26/27$.

    Also: This function assumes a minimum value of $\displaystyle \tfrac{19}{18}\approx 0.235$ at $\displaystyle x=\tfrac{\pi}{2}+n\cdot\pi, n\in\mathbb{Z}$. But it is always $\displaystyle >0$.
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  8. #8
    Super Member Failure's Avatar
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    Quote Originally Posted by pacman View Post
    and this one, (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27; what does it mean?
    It means that this time you have plotted $\displaystyle y=(16/81)^{sin^2 x} + (8/9)^{cos^2 x}$ for a larger range of x and, for some reason that I cannot fathom, seem to have surprised yourself with the result.
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  9. #9
    Senior Member pacman's Avatar
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    your observation is quite alright with me, thanks
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